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Lehmer matrix

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In mathematics, particularly matrix theory, the n×n Lehmer matrix (named after Derrick Henry Lehmer) is the constant symmetric matrix defined by

Contents

A i j = { i / j , j ≥ i j / i , j < i .

Alternatively, this may be written as

A i j = min ( i , j ) max ( i , j ) .

Properties

As can be seen in the examples section, if A is an n×n Lehmer matrix and B is an m×m Lehmer matrix, then A is a submatrix of B whenever m>n. The values of elements diminish toward zero away from the diagonal, where all elements have value 1.

The inverse of a Lehmer matrix is a tridiagonal matrix, where the superdiagonal and subdiagonal have strictly negative entries. Consider again the n×n A and m×m B Lehmer matrices, where m>n. A rather peculiar property of their inverses is that A−1 is nearly a submatrix of B−1, except for the An,n element, which is not equal to Bm,m.

A Lehmer matrix of order n has trace n.

Examples

The 2×2, 3×3 and 4×4 Lehmer matrices and their inverses are shown below.

A 2 = ( 1 1 / 2 1 / 2 1 ) ; A 2 − 1 = ( 4 / 3 − 2 / 3 − 2 / 3 4 / 3 ) ; A 3 = ( 1 1 / 2 1 / 3 1 / 2 1 2 / 3 1 / 3 2 / 3 1 ) ; A 3 − 1 = ( 4 / 3 − 2 / 3 − 2 / 3 32 / 15 − 6 / 5 − 6 / 5 9 / 5 ) ; A 4 = ( 1 1 / 2 1 / 3 1 / 4 1 / 2 1 2 / 3 1 / 2 1 / 3 2 / 3 1 3 / 4 1 / 4 1 / 2 3 / 4 1 ) ; A 4 − 1 = ( 4 / 3 − 2 / 3 − 2 / 3 32 / 15 − 6 / 5 − 6 / 5 108 / 35 − 12 / 7 − 12 / 7 16 / 7 ) .

References

Lehmer matrix Wikipedia
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